Consider a rigid body with dissimilar principal moments of inertia, satisfying I1 > 12 > 13. In the absence of torques, the components of the angular velocity at the base of the body are obtained from Euler's equations اننا 12 - 13 I₁ -W2W3 13 - 1₁ 12 -W3W1 1₁ - 12 13 -W1W2 and the orientation of the principal axes from the equation ể; = w × êį, which gives rise to the system of equations: e₁ = w3e2 - w₂e3, e2 = Wiê3 - w3e1, e3 = w₂ế1 - wiệ₂ Question/Problem (b) Consider now a small perturbation of each of these stationary solutions, of the form wi(t) = wê, +8w(t) where there is no summation convention and where Sw(t) is perpendicular to ê,. Substituting into the Euler equations and retaining terms of up to first order in do(t), find the dynamic solution for the perturbation 8(t) to linear order at the initial condition &@(0). Use your result to show that the stationary solutions w₁(t) and w₂(t) are stable but that the solution @₂(t) is unstable.

Consider a rigid body with dissimilar principal moments of inertia, satisfying I1 > 12 > 13. In the absence of torques, the components of the angular velocity at the base of the body are obtained from Euler's equations اننا 12 - 13 I₁ -W2W3 13 - 1₁ 12 -W3W1 1₁ - 12 13 -W1W2 and the orientation of the principal axes from the equation ể; = w × êį, which gives rise to the system of equations: e₁ = w3e2 - w₂e3, e2 = Wiê3 - w3e1, e3 = w₂ế1 - wiệ₂ Question/Problem (b) Consider now a small perturbation of each of these stationary solutions, of the form wi(t) = wê, +8w(t) where there is no summation convention and where Sw(t) is perpendicular to ê,. Substituting into the Euler equations and retaining terms of up to first order in do(t), find the dynamic solution for the perturbation 8(t) to linear order at the initial condition &@(0). Use your result to show that the stationary solutions w₁(t) and w₂(t) are stable but that the solution @₂(t) is unstable.